Generalized \(C(\alpha )\) Tests for Estimating Functions with Serial Dependence

  • Jean-Marie DufourEmail author
  • Alain Trognon
  • Purevdorj Tuvaandorj
Part of the Fields Institute Communications book series (FIC, volume 78)


We propose generalized \(C(\alpha )\) tests for testing linear and nonlinear parameter restrictions in models specified by estimating functions. The proposed procedures allow for general forms of serial dependence and heteroskedasticity, and can be implemented using any root-n consistent restricted estimator. The asymptotic distribution of the proposed statistic is established under weak regularity conditions. We show that earlier \(C(\alpha )\)-type statistics are included as special cases. The problem of testing hypotheses fixing a subvector of the complete parameter vector is discussed in detail as another special case. We also show that such tests provide a simple general solution to the problem of accounting for estimated parameters in the context of two-step procedures where a subvector of model parameters is estimated in a first step and then treated as fixed.


Testing \(C(\alpha )\) test Estimating function Generalized method of moment (GMM) Serial dependence Pseudo-likelihood M-estimator Nonlinear model Score test Lagrange multiplier test Heteroskedasticity 



The authors thank Marine Carrasco, Jean-Pierre Cotton, Russell Davidson, Abdeljelil Farhat, V. P. Godambe, Christian Genest, Christian Gouriéroux, Stéphane Grégoir, Tianyu He, Frank Kleibergen, Sophocles Mavroeidis, Hervé Mignon, Julien Neves, Denis Pelletier, Mohamed Taamouti, Masaya Takano, Pascale Valéry, two anonymous referees, and the Editor Wai Keung for several useful comments. Earlier versions of this paper were presented at the Canadian Statistical Society 1997 annual meeting and at INSEE (CREST, Paris). This work was supported by the William Dow Chair in Political Economy (McGill University), the Bank of Canada (Research Fellowship), the Toulouse School of Economics (Pierre-de-Fermat Chair of excellence), the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation, Germany), the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology and Complex Systems (MITACS)], the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche sur la société et la culture (Québec).


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Jean-Marie Dufour
    • 1
    • 2
    Email author
  • Alain Trognon
    • 3
    • 4
  • Purevdorj Tuvaandorj
    • 5
  1. 1.Department of EconomicsMcGill UniversityMontréalCanada
  2. 2.Centre Interuniversitaire de Recherche en Économie quantitative (CIREQ) and Centre Interuniversitaire de Recherche en Analyse des Organisations (CIRANO)MontréalCanada
  3. 3.CREST-ENSAE (Centre de Recherche en Économie et Statistique)CREST-PARISMalakoff CedexFrance
  4. 4.University Paris 1ParisFrance
  5. 5.CREST-ENSAIBruz CedexFrance

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